As regards its space it would be infinite. But calculation
shows that in a quasi-Euclidean universe the average density of matter
would necessarily be nil. Thus such a universe could not be inhabited
by matter everywhere ; it would present to us that unsatisfactory
picture which we portrayed in Section 30.
If we are to have in the universe an average density of matter which
differs from zero, however small may be that difference, then the
universe cannot be quasi-Euclidean. On the contrary, the results of
calculation indicate that if matter be distributed uniformly, the
universe would necessarily be spherical (or elliptical). Since in
reality the detailed distribution of matter is not uniform, the real
universe will deviate in individual parts from the spherical, i.e. the
universe will be quasi-spherical. But it will be necessarily finite.
In fact, the theory supplies us with a simple connection * between
the space-expanse of the universe and the average density of matter in
it.
Notes
*) For the radius R of the universe we obtain the equation
eq. 28: file eq28.gif
The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27;
p is the average density of the matter and k is a constant connected
with the Newtonian constant of gravitation.
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